{"id":1404,"date":"2025-07-25T01:03:01","date_gmt":"2025-07-25T01:03:01","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1404"},"modified":"2026-03-18T05:29:38","modified_gmt":"2026-03-18T05:29:38","slug":"complex-numbers-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/complex-numbers-fresh-take\/","title":{"raw":"Complex Numbers: Fresh Take","rendered":"Complex Numbers: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Express square roots of negative numbers as multiples of i.<\/li>\r\n \t<li>Plot complex numbers on the complex plane.<\/li>\r\n \t<li>Add, subtract, and multiply complex numbers.<\/li>\r\n \t<li>Rationalize complex denominators<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Complex Numbers<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\n<strong>Complex numbers<\/strong> are a type of number that expand the traditional notion of numbers by including imaginary numbers.\r\n\r\nThe <strong>imaginary number<\/strong> [latex]i[\/latex] is defined to be [latex]i=\\sqrt{-1}[\/latex]. Any real multiple of [latex]i[\/latex], like 5[latex]i[\/latex], is also an imaginary number.\r\n\r\nA complex number is composed of two parts: a <strong>real part<\/strong> and an <strong>imaginary part<\/strong>, often written in the form [latex]a + bi[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are real numbers. This expanded number system allows for solutions to equations that cannot be solved using only real numbers.\r\n\r\n<\/div>\r\n<section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dgdcdhac-SP-YJe7Vldo\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/SP-YJe7Vldo?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dgdcdhac-SP-YJe7Vldo\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12845025&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dgdcdhac-SP-YJe7Vldo&amp;vembed=0&amp;video_id=SP-YJe7Vldo&amp;video_target=tpm-plugin-dgdcdhac-SP-YJe7Vldo\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Introduction+to+complex+numbers+%7C+Imaginary+and+complex+numbers+%7C+Precalculus+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to complex numbers | Imaginary and complex numbers | Precalculus | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h3>Complex Plane<\/h3>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\nIn the <strong>complex plane<\/strong>, the horizontal axis is the real axis and the vertical axis is the imaginary axis.\r\n\r\nComplex Plane Structure:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Horizontal axis: Real part<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical axis: Imaginary part<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]z = a + bi[\/latex] is plotted as point [latex](a, b)[\/latex]<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_1729\" align=\"aligncenter\" width=\"205\"]<img class=\"wp-image-1729\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23184017\/axisofimaginaryreal.png\" alt=\"The vertical axis is imaginary, and the horizontal axis is real.\" width=\"205\" height=\"143\" \/> Imaginary-real graph[\/caption]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bbadchec-kGzXIbauGQk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/kGzXIbauGQk?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bbadchec-kGzXIbauGQk\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12845026&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bbadchec-kGzXIbauGQk&amp;vembed=0&amp;video_id=kGzXIbauGQk&amp;video_target=tpm-plugin-bbadchec-kGzXIbauGQk\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Plotting+complex+numbers+on+the+complex+plane+%7C+Precalculus+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPlotting complex numbers on the complex plane | Precalculus | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h3>Arithmetic on Complex Numbers<\/h3>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Addition and Subtraction:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Add\/subtract real and imaginary parts separately<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\left(a+bi\\right)+\\left(c+di\\right)=\\left(a+c\\right)+\\left(b+d\\right)i[\/latex]<\/li>\r\n \t<li>[latex]\\left(a+bi\\right)-\\left(c+di\\right)=\\left(a-c\\right)+\\left(b-d\\right)i[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiplication:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">With real number: Distribute<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Between complex numbers: Use FOIL or distributive property<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key rule: [latex]i^2 = -1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Geometric Interpretation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Addition\/Subtraction: Translation in complex plane<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiplication: Scaling and rotation<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Complex Number Division:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Cannot divide by imaginary numbers directly<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Goal: Eliminate imaginary part in denominator<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Complex Conjugate:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Definition: For [latex]a+bi[\/latex], conjugate is [latex]a-bi[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key property: [latex]\\begin{align}(a+bi)(a-bi)&amp;=a^2+b^2\\end{align}[\/latex]\u00a0(always real)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Division Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Multiply numerator and denominator by denominator's conjugate<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify and rationalize the denominator<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fgdagdgf-vPZAW7Lhh1E\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/vPZAW7Lhh1E?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fgdagdgf-vPZAW7Lhh1E\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12845027&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fgdagdgf-vPZAW7Lhh1E&amp;vembed=0&amp;video_id=vPZAW7Lhh1E&amp;video_target=tpm-plugin-fgdagdgf-vPZAW7Lhh1E\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Visualizing+complex+arithmetic_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cVisualizing complex arithmetic\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox example\">Multiply: [latex]4\\left(2+5i\\right)[\/latex].\r\n[reveal-answer q=\"703380\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"703380\"]\r\nTo multiply the complex number by a real number, we simply distribute as we would when multiplying polynomials. Distribute and simplify.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}&amp;4(2+5i)\\\\&amp;=4\\cdot2+4\\cdot5i\\\\&amp;=8+20i\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">Visualize the product [latex]i\\left(1+2i\\right)[\/latex].\r\n[reveal-answer q=\"703386\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"703386\"]\r\nMultiplying, we\u2019d get\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;i\\cdot1+i\\cdot2i\\\\&amp;=i+2{{i}^{2}}\\\\&amp;=i+2(-1)\\\\&amp;=-2+i\\\\\\end{align}[\/latex]<\/p>\r\nIn this case, the distance from the origin has not changed, but the point has been rotated about the origin, [latex]90\u00b0[\/latex] counter-clockwise.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_1735\" align=\"aligncenter\" width=\"333\"]<img class=\"wp-image-1735 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23220204\/Screen-Shot-2017-02-23-at-2.00.11-PM.png\" alt=\"The imaginary-real graph with the point 1,2, which is labeled 1 plus 2i, and the point negative 2, 1, which is labeled negative 2 plus i. A dotted red line extends from the origin to 1 plus 2i. A red arrow indicates this dotted line moves so that it extends from the origin to negative 2 plus i.\" width=\"333\" height=\"250\" \/> Imaginary-real graph demonstrating rotation[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hhacaahb-XBJjbJAwM1c\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/XBJjbJAwM1c?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hhacaahb-XBJjbJAwM1c\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12845028&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hhacaahb-XBJjbJAwM1c&amp;vembed=0&amp;video_id=XBJjbJAwM1c&amp;video_target=tpm-plugin-hhacaahb-XBJjbJAwM1c\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Dividing+Complex+Numbers_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Dividing Complex Numbers\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Complex\u00a0Roots<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Quadratic Equation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">General form: [latex]ax^2 + bx + c = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Roots are x-intercepts of the parabola<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Quadratic Formula:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solves all quadratic equations<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Complex Roots:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Occur when [latex]b^2 - 4ac[\/latex] is negative<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Always come in conjugate pairs: [latex]a + bi[\/latex] and [latex]a - bi[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">No real [latex]x[\/latex]-intercepts for the parabola<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Geometric Interpretation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Complex roots: Parabola doesn't cross x-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Real roots: Parabola touches or crosses x-axis<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Discriminant:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Expression under the square root: [latex]b^2 - 4ac[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determines nature and number of solutions<\/li>\r\n \t<li>\r\n<table summary=\"A table with 5 rows and 2 columns. The entries in the first row are: Value of Discriminant and Results. The entries in the second row are: b squared minus four times a times c equals zero and One rational solution (double solution). The entries in the third row are: b squared minus four times a times c is greater than zero, perfect square and Two rational solutions. The entries in the fourth row are: b squared minus four times a times c is greater than zero, not a perfect square and Two irrational solutions. The entries in the fifth row are: b squared minus four times a times c is less than zero and Two complex solutions.\">\r\n<thead>\r\n<tr>\r\n<th>Value of Discriminant<\/th>\r\n<th>Results<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]{b}^{2}-4ac=0[\/latex]<\/td>\r\n<td>One rational solution (double solution)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{b}^{2}-4ac&gt;0[\/latex], perfect square<\/td>\r\n<td>Two rational solutions<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{b}^{2}-4ac&gt;0[\/latex], not a perfect square<\/td>\r\n<td>Two irrational solutions<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{b}^{2}-4ac&lt;0[\/latex]<\/td>\r\n<td>Two complex solutions<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Types of Solutions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]b^2 - 4ac &gt; 0[\/latex]: Two real solutions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]b^2 - 4ac = 0[\/latex]: One real solution (double root)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]b^2 - 4ac &lt; 0[\/latex]: Two complex solutions<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cbgfbhhe-11EwTcRMPn8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/11EwTcRMPn8?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cbgfbhhe-11EwTcRMPn8\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12845029&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cbgfbhhe-11EwTcRMPn8&amp;vembed=0&amp;video_id=11EwTcRMPn8&amp;video_target=tpm-plugin-cbgfbhhe-11EwTcRMPn8\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Quadratic+Formula+-+Complex+Solutions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Quadratic Formula - Complex Solutions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Analyze the following quadratic equations using the discriminant. Determine the nature of their solutions without solving the equations:<\/p>\r\n\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li class=\"whitespace-pre-wrap break-words\">[latex]2x^2 + 5x - 3 = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-pre-wrap break-words\">[latex]x^2 - 6x + 9 = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-pre-wrap break-words\">[latex]3x^2 + 2x + 5 = 0[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"476275\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"476275\"]Recall: For a quadratic equation [latex]ax^2 + bx + c = 0[\/latex], the discriminant is [latex]b^2 - 4ac[\/latex].\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li>Identify coefficients: [latex]a = 2, b = 5, c = -3[\/latex]<\/li>\r\n \t<li>Calculate discriminant:\r\n[latex]\\begin{align*} b^2 - 4ac &amp;= 5^2 - 4(2)(-3) \\\\ &amp;= 25 - (-24) \\\\ &amp;= 25 + 24 \\\\ &amp;= 49 \\end{align*}[\/latex]<\/li>\r\n \t<li>Interpret:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]49 &gt; 0[\/latex] and is a perfect square<\/li>\r\n \t<li class=\"whitespace-normal break-words\">This equation has two distinct rational solutions<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li>Identify coefficients: [latex]a = 1, b = -6, c = 9[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate discriminant:\r\n[latex]\\begin{align*} b^2 - 4ac &amp;= (-6)^2 - 4(1)(9) \\\\ &amp;= 36 - 36 \\\\ &amp;= 0 \\end{align*}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Interpret:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]0 = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">This equation has one real solution (a double root)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li>Identify coefficients: [latex]a = 3, b = 2, c = 5[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate discriminant:\r\n[latex]\\begin{align*} b^2 - 4ac &amp;= 2^2 - 4(3)(5) \\\\ &amp;= 4 - 60 \\\\ &amp;= -56 \\end{align*}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Interpret:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]-56 &lt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">This equation has two complex solutions<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>&nbsp;","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Express square roots of negative numbers as multiples of i.<\/li>\n<li>Plot complex numbers on the complex plane.<\/li>\n<li>Add, subtract, and multiply complex numbers.<\/li>\n<li>Rationalize complex denominators<\/li>\n<\/ul>\n<\/section>\n<h2>Complex Numbers<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Complex numbers<\/strong> are a type of number that expand the traditional notion of numbers by including imaginary numbers.<\/p>\n<p>The <strong>imaginary number<\/strong> [latex]i[\/latex] is defined to be [latex]i=\\sqrt{-1}[\/latex]. Any real multiple of [latex]i[\/latex], like 5[latex]i[\/latex], is also an imaginary number.<\/p>\n<p>A complex number is composed of two parts: a <strong>real part<\/strong> and an <strong>imaginary part<\/strong>, often written in the form [latex]a + bi[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are real numbers. This expanded number system allows for solutions to equations that cannot be solved using only real numbers.<\/p>\n<\/div>\n<section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dgdcdhac-SP-YJe7Vldo\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/SP-YJe7Vldo?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dgdcdhac-SP-YJe7Vldo\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12845025&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dgdcdhac-SP-YJe7Vldo&amp;vembed=0&amp;video_id=SP-YJe7Vldo&amp;video_target=tpm-plugin-dgdcdhac-SP-YJe7Vldo\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Introduction+to+complex+numbers+%7C+Imaginary+and+complex+numbers+%7C+Precalculus+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to complex numbers | Imaginary and complex numbers | Precalculus | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Complex Plane<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>In the <strong>complex plane<\/strong>, the horizontal axis is the real axis and the vertical axis is the imaginary axis.<\/p>\n<p>Complex Plane Structure:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Horizontal axis: Real part<\/li>\n<li class=\"whitespace-normal break-words\">Vertical axis: Imaginary part<\/li>\n<li class=\"whitespace-normal break-words\">[latex]z = a + bi[\/latex] is plotted as point [latex](a, b)[\/latex]<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_1729\" aria-describedby=\"caption-attachment-1729\" style=\"width: 205px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1729\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23184017\/axisofimaginaryreal.png\" alt=\"The vertical axis is imaginary, and the horizontal axis is real.\" width=\"205\" height=\"143\" \/><figcaption id=\"caption-attachment-1729\" class=\"wp-caption-text\">Imaginary-real graph<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<\/div>\n<section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bbadchec-kGzXIbauGQk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/kGzXIbauGQk?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bbadchec-kGzXIbauGQk\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12845026&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bbadchec-kGzXIbauGQk&amp;vembed=0&amp;video_id=kGzXIbauGQk&amp;video_target=tpm-plugin-bbadchec-kGzXIbauGQk\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Plotting+complex+numbers+on+the+complex+plane+%7C+Precalculus+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPlotting complex numbers on the complex plane | Precalculus | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Arithmetic on Complex Numbers<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Addition and Subtraction:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Add\/subtract real and imaginary parts separately<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\left(a+bi\\right)+\\left(c+di\\right)=\\left(a+c\\right)+\\left(b+d\\right)i[\/latex]<\/li>\n<li>[latex]\\left(a+bi\\right)-\\left(c+di\\right)=\\left(a-c\\right)+\\left(b-d\\right)i[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Multiplication:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">With real number: Distribute<\/li>\n<li class=\"whitespace-normal break-words\">Between complex numbers: Use FOIL or distributive property<\/li>\n<li class=\"whitespace-normal break-words\">Key rule: [latex]i^2 = -1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Geometric Interpretation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Addition\/Subtraction: Translation in complex plane<\/li>\n<li class=\"whitespace-normal break-words\">Multiplication: Scaling and rotation<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Complex Number Division:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Cannot divide by imaginary numbers directly<\/li>\n<li class=\"whitespace-normal break-words\">Goal: Eliminate imaginary part in denominator<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Complex Conjugate:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Definition: For [latex]a+bi[\/latex], conjugate is [latex]a-bi[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Key property: [latex]\\begin{align}(a+bi)(a-bi)&=a^2+b^2\\end{align}[\/latex]\u00a0(always real)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Division Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Multiply numerator and denominator by denominator&#8217;s conjugate<\/li>\n<li class=\"whitespace-normal break-words\">Simplify and rationalize the denominator<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fgdagdgf-vPZAW7Lhh1E\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/vPZAW7Lhh1E?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fgdagdgf-vPZAW7Lhh1E\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12845027&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fgdagdgf-vPZAW7Lhh1E&amp;vembed=0&amp;video_id=vPZAW7Lhh1E&amp;video_target=tpm-plugin-fgdagdgf-vPZAW7Lhh1E\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Visualizing+complex+arithmetic_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cVisualizing complex arithmetic\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">Multiply: [latex]4\\left(2+5i\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q703380\">Show Solution<\/button><\/p>\n<div id=\"q703380\" class=\"hidden-answer\" style=\"display: none\">\nTo multiply the complex number by a real number, we simply distribute as we would when multiplying polynomials. Distribute and simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}&4(2+5i)\\\\&=4\\cdot2+4\\cdot5i\\\\&=8+20i\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Visualize the product [latex]i\\left(1+2i\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q703386\">Show Solution<\/button><\/p>\n<div id=\"q703386\" class=\"hidden-answer\" style=\"display: none\">\nMultiplying, we\u2019d get<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&i\\cdot1+i\\cdot2i\\\\&=i+2{{i}^{2}}\\\\&=i+2(-1)\\\\&=-2+i\\\\\\end{align}[\/latex]<\/p>\n<p>In this case, the distance from the origin has not changed, but the point has been rotated about the origin, [latex]90\u00b0[\/latex] counter-clockwise.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_1735\" aria-describedby=\"caption-attachment-1735\" style=\"width: 333px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1735 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23220204\/Screen-Shot-2017-02-23-at-2.00.11-PM.png\" alt=\"The imaginary-real graph with the point 1,2, which is labeled 1 plus 2i, and the point negative 2, 1, which is labeled negative 2 plus i. A dotted red line extends from the origin to 1 plus 2i. A red arrow indicates this dotted line moves so that it extends from the origin to negative 2 plus i.\" width=\"333\" height=\"250\" \/><figcaption id=\"caption-attachment-1735\" class=\"wp-caption-text\">Imaginary-real graph demonstrating rotation<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hhacaahb-XBJjbJAwM1c\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/XBJjbJAwM1c?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hhacaahb-XBJjbJAwM1c\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12845028&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hhacaahb-XBJjbJAwM1c&amp;vembed=0&amp;video_id=XBJjbJAwM1c&amp;video_target=tpm-plugin-hhacaahb-XBJjbJAwM1c\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Dividing+Complex+Numbers_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Dividing Complex Numbers\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Complex\u00a0Roots<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Quadratic Equation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">General form: [latex]ax^2 + bx + c = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Roots are x-intercepts of the parabola<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Quadratic Formula:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Solves all quadratic equations<\/li>\n<\/ul>\n<\/li>\n<li>Complex Roots:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Occur when [latex]b^2 - 4ac[\/latex] is negative<\/li>\n<li class=\"whitespace-normal break-words\">Always come in conjugate pairs: [latex]a + bi[\/latex] and [latex]a - bi[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">No real [latex]x[\/latex]-intercepts for the parabola<\/li>\n<\/ul>\n<\/li>\n<li>Geometric Interpretation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Complex roots: Parabola doesn&#8217;t cross x-axis<\/li>\n<li class=\"whitespace-normal break-words\">Real roots: Parabola touches or crosses x-axis<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ul>\n<li class=\"whitespace-normal break-words\">Discriminant:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Expression under the square root: [latex]b^2 - 4ac[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Determines nature and number of solutions<\/li>\n<li>\n<table summary=\"A table with 5 rows and 2 columns. The entries in the first row are: Value of Discriminant and Results. The entries in the second row are: b squared minus four times a times c equals zero and One rational solution (double solution). The entries in the third row are: b squared minus four times a times c is greater than zero, perfect square and Two rational solutions. The entries in the fourth row are: b squared minus four times a times c is greater than zero, not a perfect square and Two irrational solutions. The entries in the fifth row are: b squared minus four times a times c is less than zero and Two complex solutions.\">\n<thead>\n<tr>\n<th>Value of Discriminant<\/th>\n<th>Results<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]{b}^{2}-4ac=0[\/latex]<\/td>\n<td>One rational solution (double solution)<\/td>\n<\/tr>\n<tr>\n<td>[latex]{b}^{2}-4ac>0[\/latex], perfect square<\/td>\n<td>Two rational solutions<\/td>\n<\/tr>\n<tr>\n<td>[latex]{b}^{2}-4ac>0[\/latex], not a perfect square<\/td>\n<td>Two irrational solutions<\/td>\n<\/tr>\n<tr>\n<td>[latex]{b}^{2}-4ac<0[\/latex]<\/td>\n<td>Two complex solutions<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Types of Solutions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]b^2 - 4ac > 0[\/latex]: Two real solutions<\/li>\n<li class=\"whitespace-normal break-words\">[latex]b^2 - 4ac = 0[\/latex]: One real solution (double root)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]b^2 - 4ac < 0[\/latex]: Two complex solutions<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cbgfbhhe-11EwTcRMPn8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/11EwTcRMPn8?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cbgfbhhe-11EwTcRMPn8\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12845029&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cbgfbhhe-11EwTcRMPn8&amp;vembed=0&amp;video_id=11EwTcRMPn8&amp;video_target=tpm-plugin-cbgfbhhe-11EwTcRMPn8\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Quadratic+Formula+-+Complex+Solutions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Quadratic Formula &#8211; Complex Solutions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Analyze the following quadratic equations using the discriminant. Determine the nature of their solutions without solving the equations:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li class=\"whitespace-pre-wrap break-words\">[latex]2x^2 + 5x - 3 = 0[\/latex]<\/li>\n<li class=\"whitespace-pre-wrap break-words\">[latex]x^2 - 6x + 9 = 0[\/latex]<\/li>\n<li class=\"whitespace-pre-wrap break-words\">[latex]3x^2 + 2x + 5 = 0[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q476275\">Show Answer<\/button><\/p>\n<div id=\"q476275\" class=\"hidden-answer\" style=\"display: none\">Recall: For a quadratic equation [latex]ax^2 + bx + c = 0[\/latex], the discriminant is [latex]b^2 - 4ac[\/latex].<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>\n<ol style=\"list-style-type: decimal;\">\n<li>Identify coefficients: [latex]a = 2, b = 5, c = -3[\/latex]<\/li>\n<li>Calculate discriminant:<br \/>\n[latex]\\begin{align*} b^2 - 4ac &= 5^2 - 4(2)(-3) \\\\ &= 25 - (-24) \\\\ &= 25 + 24 \\\\ &= 49 \\end{align*}[\/latex]<\/li>\n<li>Interpret:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]49 > 0[\/latex] and is a perfect square<\/li>\n<li class=\"whitespace-normal break-words\">This equation has two distinct rational solutions<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/li>\n<li class=\"whitespace-normal break-words\">\n<ol style=\"list-style-type: decimal;\">\n<li>Identify coefficients: [latex]a = 1, b = -6, c = 9[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Calculate discriminant:<br \/>\n[latex]\\begin{align*} b^2 - 4ac &= (-6)^2 - 4(1)(9) \\\\ &= 36 - 36 \\\\ &= 0 \\end{align*}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Interpret:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]0 = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">This equation has one real solution (a double root)<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/li>\n<li class=\"whitespace-normal break-words\">\n<ol style=\"list-style-type: decimal;\">\n<li>Identify coefficients: [latex]a = 3, b = 2, c = 5[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Calculate discriminant:<br \/>\n[latex]\\begin{align*} b^2 - 4ac &= 2^2 - 4(3)(5) \\\\ &= 4 - 60 \\\\ &= -56 \\end{align*}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Interpret:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]-56 < 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">This equation has two complex solutions<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>&nbsp;<\/p>\n","protected":false},"author":67,"menu_order":16,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Introduction to complex numbers | Imaginary and complex numbers | Precalculus | Khan Academy\",\"author\":\"\",\"organization\":\"Khan Academy\",\"url\":\"https:\/\/youtu.be\/SP-YJe7Vldo\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Plotting complex numbers on the complex plane | Precalculus | Khan Academy\",\"author\":\"\",\"organization\":\"Khan Academy\",\"url\":\"https:\/\/youtu.be\/kGzXIbauGQk\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Visualizing complex arithmetic\",\"author\":\"\",\"organization\":\"OCLPhase2\",\"url\":\"https:\/\/youtu.be\/vPZAW7Lhh1E\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Dividing Complex Numbers\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/XBJjbJAwM1c\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Quadratic Formula - 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